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Computers and Chemical Engineering 31 (2007) 1222–1230

Control of a batch pulp digester using a simplified mechanisticmodel to predict degree of polymerisation

Philip de Vaal ∗, Carl SandrockDepartment of Chemical Engineering, University of Pretoria, Pretoria 0001, South Africa

Received 31 March 2005; received in revised form 23 June 2006; accepted 12 October 2006Available online 13 December 2006

bstract

Effective control of the sulphite pulp digestion process in the production of dissolved pulp in a batch digester is limited by three importantestrictions, namely:

the inability to measure the degree of polymerisation (DP) of the cellulose in the wood pulp, which is the controlled variable,the fact that flowrate of steam to an external heat exchanger, through which the cooking liquor circulates, is the only manipulated variableavailable and that,due to scheduling requirements, cook time per digester is fixed.

The availability of substantial computational capability in industrial environments has rendered the traditional S-factor prediction of cook timeo control DP inadequate, compared to what can be achieved with more sophisticated predictive models. A fundamental model, in simplified form,ith adjustable parameters, was developed and its accuracy to predict DP based on given operating conditions was tested using available plantata. This model was built into a control structure for implementation on an operational batch digester.

Because the parameters form part of fundamental relationships in the model, realistic bounds can be placed on them during the optimisationrocess, enabling a better understanding of the behaviour of the model.

Overall control of the batch digestion process takes place in three interdependent, but distinct time frames, namely continuous control of digester

emperature based on a calculated setpoint, using a model-based approach to predict DP on a cook-to-cook basis and lastly while the final DP muste kept on target through a number of cooks (N cooks), taking into account long-term changes in the process. This long-term optimisation takeslace in the third time frame.

2006 Elsevier Ltd. All rights reserved.

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eywords: Automatic tuning; Performance; Pulp digester; Control; Batch

. Introduction

In the production of dissolved pulp, special emphasis islaced on the quality of the pulp, especially with respect to fibreength. Unlike many other processes, where back-blending ofroduct can ensure adherence to product quality requirements,his is not possible in the case of dissolved pulp production.

lthough modern pulp manufacturing plants make use of contin-ous pulping digesters, several large plants are still operational,here batch digesters are used for production of dissolved pulp.

∗ Corresponding author. Tel.: +27 12 420 2475; fax: +27 12 420 5048.E-mail address: philip.devaal@up.ac.za (P. de Vaal).

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098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2006.10.015

o enable production targets, the control challenges on such alant require innovative approaches to ensure sustained trouble-ree operation.

The control challenges to be addressed are:

Each digester operates on a batch cycle over an operatingrange.The controlled variable for quality, the degree of degradationof the pulp, is not continuously measurable.

Only one manipulated variable, steam supply rate, is avail-able.In order to ensure continuous supply of pulp to the down-stream processes, strict requirements exist for each digester

Chemical Engineering 31 (2007) 1222–1230 1223

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P. de Vaal, C. Sandrock / Computers and

with respect to available time within which to completecooks.The controller has to operate within specific constraints,which include limits on available steam and continu-ously varying heat transfer area in addition to unmeasureddisturbances associated with liquor strength, wood chip com-position, moisture content, etc.

The conventional approach towards control of pulp viscositys to estimate cook duration based on a prediction of requirednergy which is in turn based on a simplified reaction rate forelignification of the wood chips. This methodology is com-only known as the S-factor approach, in the case of dissolved

ulp production.In order to achieve tighter control of pulp viscosity under

he stated specifications, a mechanistic model, with adjustablearameters was used to achieve improved control of pulp viscos-ty. In this paper it is shown that substantially improved controls possible by making use of a more sophisticated mechanistic

odel, and applying optimisation techniques to adjust modelarameters, ensuring tighter control.

. Traditional control of pulp viscosity

The SAPPI SAICCOR plant in Umkomaas, KwaZulu-Nataln South Africa produces dissolving pulp for the production of

wide range of products. Acid sulphite pulping in 23 batchigesters is used to produce the pulp from mainly Eucalyp-us wood. Since dissolving pulps are produced, the aim of therocess is to produce pulp with a specified final degree of poly-erisation.Temperature in the digester is used to control the cellulose

egradation, while the steam flow to the external heat exchangerss used to control digester temperature in a cascaded arrangements shown in Fig. 1.

During the manufacture of dissolving pulps, the controlledariable is the degree of polymerisation (DP) of the cellulose,hich gives an indication of the average length of the polymer

hains, but which cannot be measured directly. DP is stronglyffected by temperature and pH.

pH is rarely measured on a batch plant, because of the costf installing and maintaining pH probes on the large number ofndividual digesters. The DP of pulp is related to the intrinsiciscosity and can be determined by measuring the viscosity ofpulp sample. This method is simple and relatively quick and

s most commonly used for the estimation of the DP value ofhe cellulose. The correlation between the DP value and the

easured viscosity was determined by Watson (1992) and wasound to be:

P = 285.4μ0.345 (1)

his correlation is valid over a range of 20 cP < μ < 150 cP.

Control of temperature is accurate, indicating that the con-

rol problem is associated with the difficulty in determining theemperature setpoint profile for the temperature controller. A

odified S-factor model, which is currently in use at SAIC-

mtSo

Fig. 1. Two-way control strategy per digester.

OR relates the degree of polymerisation to temperature in theigester in order to be able to calculate the temperature setpointshat will ensure obtaining the desired DP in the available cookime.

The S-factor model was originally developed by Yorston andiebergott (1965) and was based on the assumption that a corre-

ation exists between the lignin content of the solid phase in theeaction and the pulp viscosity. The widely accepted delignifica-ion rate equation, in the form also later reported by Hagberg andchoon (1973), was used to model delignification and relate theulp viscosity to that. The Arrhenius temperature dependencyf the reaction had already been confirmed and this was used inhe following equation:

F = −∫ Lf

L0

1

A0[L]d[L] =

∫ tf

t0

e{B−(E0/RT )}dt (2)

ntegration of the left-hand side of this equation yields:

Lf] = k1 e{−A0 SF} (3)

From the initial assumption, the residual lignin concentrationas taken to be proportional to the cuprammonium viscosity of

he pulp, leading to the following:

= k2 e{−A0 SF} (4)

he parameters k1 and k2 are constant. Eq. (4) formed the basisf the “S-factor” model. Individual mills adapted the S-factor

odel to their own data and observations in order to improve

he control of their respective plants, for example using the actualO2 concentration of the liquor instead of the partial pressuref the SO2 (Marr & Bondy, 1986).

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224 P. de Vaal, C. Sandrock / Computers and

The weakness of the S-factor model lies in the assumptionhat the residual lignin content is proportional to the degree ofolymerisation of the cellulose.

The basis for any new control strategy would be a modelhat can relate the degree of polymerisation to the manipulatedariable or variables. This will be necessary, since the degree ofolymerisation cannot be measured continuously. The digesteronditions as well as the degree of polymerisation must there-ore be simulated and the control done based on the predictionsf this simulation. It was shown (Kilian & de Vaal, 2000) thathe best variable to use for the control of the final degree ofolymerisation will be the temperature profile. That will also behe manipulated variable used in this study. A model is thereforeecessary to relate the temperature to the degree of polymerisa-ion.

. Improved model-based approach

.1. The mechanistic model

A lot of detail is necessary in a fundamental model to definehe system completely, which can make the task of fundamen-al modelling very complicated. Empirical modelling can be

good alternative to fundamental modelling if the necessaryquipment is available. It can often be performed more quicklynd requires only minimum insight into the process. As bothodelling approaches have their obvious drawbacks, they are

requently combined to a certain extent, in order to obtain mean-ngful models of limited complexity with minimum modellingffort, while still covering the operating regime of interest forhe optimisation.

The approach taken during this study for the modelling ofhe digester was to combine the empirical and fundamental mod-lling techniques. A general fundamental model was constructedrom the knowledge of the acid sulphite process and the reactionsaking place during the pulping process. This model structureontained the basic reaction mechanisms, as well as the relevantquilibrium conditions. However, one of the biggest problemsuring fundamental modelling is to obtain accurate parametersor the various reaction rate equations and the correspondingquilibrium conditions. The model was therefore created withhe ability to easily adjust the parameters. The model was thentted to experimental and actual plant data and the parametersomputed to best fit the real observations. Computer softwareas written to assist in the initial process of adjusting the param-

ters in order to obtain an accurate model.The software programme facilitates the process of construct-

ng an accurate process model over a batch pulp digester.owever, the basic model structure has to be in place alreadyefore the package can be used to fit the model to actual data.his implies that the programme doesn’t apply process iden-

ification techniques or artificial intelligence to fit a black-boxodel to the process. It is used as a tool to fine-tune the parame-

ers of an already developed model, as well as for the verificationf the model and is generally applicable to batch pulping pro-esses, and by changing the process model used, could also besed for other batch processes as well.

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ical Engineering 31 (2007) 1222–1230

Development and verification of the digester model wasivided into three parts, namely:

the kinetics of the reactions involving the wood constituents,the liquor composition, andthe pressure and the temperature inside the digester.

For verification of the model, the first two parts were consid-red together and the third part separately. The reaction kineticsnd the liquor composition were verified using data obtainedrom a pilot scale facility (Watson, 1992), since intermediateeasurements of the wood and liquor conditions were possi-

le using the laboratory equipment. Modelling of the actualonditions inside the digester were verified with real plantata. It was reasoned that if the verification of the kinetics,sing known conditions for the modelling of the reactions,hows that the kinetic model is correct and the verificationf the modelling of the conditions inside the digester giveood results, then the entire model can be assumed to beorrect.

The software programme was used to do the verification of theodel, as well as to simplify the adjustment of the parameters

n the model to obtain a model that represents the real-worldituation.

The approach taken in this study was similar to the one men-ioned by Terwiesch and Agarwal (1995). This approach requireshat the basic model structure be chosen from first-principlesonsiderations, while the model parameters are typically com-uted so that the model predictions best fit the data. The softwareherefore allows for the parameters in the model to be adjustedo that the modelling results best fit the data.

The most important variable in the model for control pur-oses, is the degree of polymerisation of the cellulose. Therimary aim is therefore the accuracy of the simulation of theegradation increase. However, it will also be necessary to ver-fy the correctness of the other parameters. This enables oneo locate the source of the largest deviation from the real val-es. Therefore, if large errors in the degree of polymerisationccur, the variable that is mainly responsible for the errors cane located. The possibility to adapt the model for the variableshat give the worst results is then created.

The correctness of the following variables were verified:

(i) Liquor samples:• bisulphite ion concentration;• pH of the liquor.

ii) Pulp samples:• lignin content;• hemicellulose content;• degree of polymerisation of the cellulose.

The simulation was done using the software package thatas created. After the simulation had been completed, the mea-

urements taken on the mini digesters were compared withhe calculated values from the simulation at the correspondingimes. A correlation coefficient was then calculated for eachariable.

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P. de Vaal, C. Sandrock / Computers and

The derivation of a simplified mechanistic model (Kilian &e Vaal, 2000), used by the controller can be summarised by theollowing assumptions:

The reactions in the digester can be accounted for by mod-elling the rate of cellulose degradation, lignin dissolutionand strong acid equilibria in the digester. The hemicellulosedegradation rate is implicitly calculated in these equations.The initial composition of the reaction mixture is known froma test of the liquor.The temperature and pressure in the digester are known.

To model the reactions, the following reaction rates are used:

d[L]

dt= kL[L]a[HSO3

−]α[H+]

β(5)

d[SA−]

dt=

{(g

v+ 2h

v

)([L]0 − [L])

}rL

+kSA(T )

ν([L]0 − [L])q[HSO3

−]b[H+]

c(6)

d[C]

dt= kC[H+]

δ(7)

he temperature dependence of all the k factors above, can beescribed by the Arrhenius relationship, k = k0 e(−E/T), with End k0 known for each of these reactions.

The hydrogen and bisulphite ion concentrations are obtainedia electroneutrality arguments combined with the assumptionhat equilibrium is attained between the SO2 in the vapour andiquid phases, as shown below:

Chemical equilibrium:

KSO2 = [H+][HSO3−]

[SO2](8)

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able 1ariables defining the digester model

ariable Symbol

esidual lignin concentration [L]ellulose concentration [C]ulfite concentration [HSO3

−]ydrogen ion concentration [H+]trong acid concentration [SA−]emperature Tressure Partial pressure of SO2 pSO2

artial pressure of water pH2O

quilibrium constant of SO2 kSO2

apour–liquid equilibrium constant Kp

ombined SO2 Combined SO2

ree SO2 Free SO2

otal positive charge [M+]iquor:wood ratio ν

ical Engineering 31 (2007) 1222–1230 1225

Vapour–liquid equilibrium:

KP = [H+][HSO3−]

pSO2

(9)

Electron neutrality:

[M+] + [H+] = [HSO3−] + [SA−] (10)

[M+] = 2[combined SO2] (11)

A further relationship required is for the total pressure of theystem:

= pSO2 + pH2O (12)

The two temperature-dependent equilibrium constants can bealculated from parameters in the shape of the Antoine equation,s follows:

og10 KSO2 = BSO2 + CSO2

T(13)

nd

og10 Kp = Bp + Cp

T(14)

n the 10 independent Eqs. (5)–(14), there are 15 variables, thiseaves five variables to be specified.

The temperature and pressure are read from the environmenturing a cook and specified by approximations during the sim-lation. At the start of a cook, the liquor-to-wood ratio is knownnd the free and combined SO2 concentrations are read duringhe 1-h test. Specifying these five variables completely specifies

he problem.

The variables defining the digester model are given in Table 1nd the associated parameters in the simplified mechanisticodel in Table 2 below.

Units Typical range

Low High

mass% 0 30kmol/m3 300 2100kmol/m3

kmol/m3

kmol/m3

K 288 423Pa 103 940Pa 700 940Pa 0 240kmol/m3

(kmol/m3)2/Pakmol/m3

kmol/m3

kmol/m3

l/kg 3:1 6:1

1226 P. de Vaal, C. Sandrock / Computers and Chem

Table 2Parameters representing the simplified mechanistic model

[L] > 12.4 [L] < 12.4

Lignin reaction kinetics (Hagberg & Schoon, 1973)k0

L (%) 1.346 × 1012 0.407 × 1012

EL (K) 12,500 12,500a 0.646 1.621α 0.819 0.765β 0.705 0.779

Cellulose reaction kinetics (Kilian & de Vaal, 2000)k0

C 7.68 × 1012

EC (K) 17,100δ 1

Strong acids reaction kinetics (Kilian & de Vaal, 2000)k0

SA (kmol/m3) 1.887 × 1016

ESA (K) 19,860q 2.2231b 1.681c 0.653g 6.87 × 10−3

h −1.310 × 10−4

Temperature range (◦C) BSO2 CSO2

Antoine coefficients (Kilian & de Vaal, 2000)KSO2

20–80 −5.077 104580–120 −6.916 1700120–150 −7.971 2133

Temperature range (◦C) Bp Cp

Antoine coefficients (Kilian & de Vaal, 2000)Kp

20–80 −9.430 2392

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.2. The model-based controller

The control objective of any controller can be seen as keepinghe effect of disturbances small while allowing the controlledystem to produce the specified outputs, as well as allowinghe system to move between desired outputs in a satisfactory

anner. This leads to a conceptual picture of the control processequired for the digester.

The controller is based on a general feedforward structure,s shown in Fig. 2. It should be clear that some form of inverseodel is required as with all feedforward control. This inverse

s obtained iteratively by obtaining the value of Ttop that givesP = DPset. This is due to the fact that a highly non-linear timeomain model has already been developed, and finding thisnverse is computationally intensive.

.3. Adaptive model

.3.1. Control time framesControl time frames are an important aspect of layered control

ystems. Control of the digester takes place in three differentime frames. The first is the continuous time frame. Control

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ical Engineering 31 (2007) 1222–1230

n this level involves keeping the top temperature (Ttop) on aemperature setpoint. Control of the final DP takes place perook, representing the second time frame, while the final DPust be kept on target through a number of cooks (N cooks),

aking into account long-term changes in the process. This long-erm optimisation takes place in the third time frame. The secondnd third time frames both encapsulate discrete control problemsith small sampling rates.In Figs. 2 and 3, increased sampling frequency is indicated

y thicker lines connecting units.

.3.2. Continuous temperature controlThis control configuration corresponds to the normal feed-

ack controller setup. The temperature in the digester isontrolled by manipulating the steam flow rate to the externaligester. This first level of control is already handled satisfacto-ily by the feedback controllers on each digester.

The data read continuously are used directly in the processodel. They are not stored during the cook process, as this was

ound to be too slow to allow real time processing. During theontrol simulation, feedback is given to the personnel using theoftware by updating graphics on the user interface showing theurrent temperatures, side circulation flow rate and pressure readrom the plant, as well as the values of the calculated pH andstimated DP.

The different phases of the control problem require separatection. The initial phase, until the 1 h test becomes available,imply stores data during this phase. Once the data becomesvailable, it is used to deduce the starting conditions of theigester.

.3.3. Per cook DP controlThe temperature setpoint for the low-level temperature con-

rol system is supplied by a profile generator which ensures thatpredefined temperature profile is followed. The profile gener-

tor requires the value of Ttop, the constant top temperate for theook, in order for it to generate a complete profile.

The final DP of the cook is determined by the temperaturerofile that the digester is taken through. Even though the profileenerator determines this profile, the value of Ttop is specifiedxternally. It should be clear that the value of Ttop thereforeetermines the value of the final DP to a great extent.

.3.4. N cook time frameThe results of a number of cooks are collected using a histo-

ian system. These stored results can be analysed to check theerformance of the system over time. New model parameters canlso be passed to the DP controller so that long-term changes inlant behaviour can be accommodated. The diagram shown inig. 2 shows the inputs and outputs of the N cook time frame.

In this time frame, the controller responsible for outputtinghe correct value of Ttop to the profile generator is shown. This ishe UP controller. The model parameters used in the controllers

re changed every N cooks, placing this update in the N cooksime frame. It should be borne in mind that the controller readsrocess variables from the plant with other sampling frequenciess well. Fig. 2 is intended to show the position of the controller

P. de Vaal, C. Sandrock / Computers and Chemical Engineering 31 (2007) 1222–1230 1227

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n the time frames, but this is by no means a unique representa-

ion.

Ultimately all control action is implemented by manipulationf the steam flow rate, but the low level control systems in placeo control the temperature profiles are working well enough so

tb

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Fig. 3. Conceptual c

e model.

hat the controller just has to concern itself with determining

he optimal top temperature that will result in the correct DPeing obtained.

After N cooks have passed (at the moment N = 10), the param-ters have to be optimised to ensure the best fit between the

ontrol scheme.

1228 P. de Vaal, C. Sandrock / Computers and Chemical Engineering 31 (2007) 1222–1230

Fig. 4. Flow diagram for the control strategy.

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P. de Vaal, C. Sandrock / Computers and

odel predictions and the real results of the previous N cooks.his is done using Matlab’s built in optimisation routines whichre based on the Simplex method.

Upon examination of Fig. 2, it is clear that the controller doesot have any specific measurement of control error. When theonditions on the plant change, causing plant-model mismatch,ontrol can be expected to degrade. In order to keep the modelarameters in line with the measured results from the plant, its necessary to provide a feedback mechanism. This is donesing a periodic optimisation of the model parameters. At aredetermined frequency, the model parameters are adapted byptimising the fit of the model to the measured DP values for theooks since the last optimisation. This ensures that the model isonstantly in close accord with the conditions on the plant. Thiss illustrated in Fig. 3.

.3.4.1. Plant interaction. Interaction with the plant waschieved using direct queries to the plant database to obtain allhe data required for operation. Both continuous and historicalata can be obtained in this manner, so that operation is entirelyutomated.

lgorithm. The controller described above is implementedsing the following algorithm:

Initial model parameters are loaded from a file. The modelparameters stored in file are listed in tables.The program reads status information from the plant, waitingfor a cook to start.The database system stores a state tag which is read to deter-mine whether a cook has started.As soon as the start of a cook has been detected, the con-troller reads plant data and stores the information internally.Note that no simulation is done in this phase, as the initialconditions for the integration of the model equations are onlyavailable after roughly an hour, once the analysis of the liquorsample as been done.As the program reads data from the plant, it checks whetherthe results of the 1-h test have become available. Thistest supplies the liquor composition at the time of thetest.When the 1-h test results are available, the controller programsimulates the cook up to the current point in time using thestored process variables and initial values calculated fromthe test results. The simulation is done by integrating thedifferential equations given in Eqs. (4)–(6) and solving theremaining equations to yield all the variables constituting thepulp degradation model.The program then simulates reactions inside the digester inreal time using the same equations, incorporating real timevalues of the process variables (temperature, pressure and theflow through the heat exchanger) that are available from theplant database system.

When the temperature reaches a predetermined temperature,TC, the program starts an optimisation routine that simulatesthe remainder of the cook using the current states as the initialconditions for each trial cook. The optimisation finds a value

ical Engineering 31 (2007) 1222–1230 1229

for Ttop that will minimise the viscosity error at the end of thecook.This temperature is displayed on the screen for entry by anoperator into the plant control system. The plant control sys-tem uses this value to ensure that the temperature is rampedup to Ttop in the required time.The previous two steps are repeated 5 min after the first pre-diction to verify the correct operation of the predictor.The model equations are integrated up to the end of the cook.In intervals of 10 cooks, the data for the last 10 cooks are readfrom the plant database, and a separate optimisation processis initiated to reduce the error between the measured DP andthe DP predicted by the model for each of these cooks.

This control strategy is shown in a flow diagram in Fig. 4.

. In situ control results

.1. Results

The UP controller was used to control 13 cooks during theeriod from May 2003 to June 2003. Fig. 5 shows the perfor-ance of the S-factor controller during the same period. The

orizontal dashed line shows an average deviation from targetlightly above 10. Note that the DP cycles dramatically betweenigh and low extremes during this time period. This is typicalf the overall operation of this controller.

Several reasons for this cyclic behaviour can be suggested.he most likely explanation is aggressive modification of theontroller parameters by plant personnel, without waiting forhe effects of these modifications to become apparent.

Fig. 6 shows the performance of the UP controller duringhe same time period. The vertical line shows the point at which

Fig. 5. Deviation from target for S-factor controller, May to June 2003.

1230 P. de Vaal, C. Sandrock / Computers and Chem

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Fig. 6. Deviation from target for UP-controller, May to June 2003.

here the same parameters are used, as was done in the figurehowing UP controller performance.

The figures show clear evidence of the improved controlerformance of the UP controller. After parameter tuning, theeviation from target was significantly lowered, and the variances consistently lower than that of the S-factor controller duringhe same time period.

.2. Variance reduction

The variance of the S-factor controller during the test waseasured as 21. The UP controller variance was 18 before

arameter tuning and at 12 after parameter tuning. Resultserived from a simulated model of the process (Sandrock &e Vaal, 2003) suggest that this figure can be expected to dropurther when the parameters are tuned again. Even if the vari-nce does not improve, the UP controller shows a 43% lowerariance than the current controller.

.3. Error reduction

The S-factor controller showed an average error of 10 units.he high average error can be explained by the fact that person-el are loathe to obtain low viscosities. Given that the S-factorontroller exhibits a high variance, average viscosities are con-istently high to avoid low viscosity results. The UP controllerhowed an average error of only one unit after parameter tuning.utomatic tuning of parameters will keep the error low.

. Conclusions

A new model-based control algorithm that interacts with anndustrial batch digester was developed and has been imple-

Y

ical Engineering 31 (2007) 1222–1230

ented on one of the batch digesters in operation. The controlredictions made by the new controller led to improved con-rol compared with the current controller which is based on a

odified S-factor model.A number of issues were identified during the course of this

nvestigation:

The model used by the UP controller does not account forall the variability observed on the plant. If these sources ofvariability can be identified, they might provide valuable cluesas to further ways of reducing the output variance.The possibility of adding additional measurements is also aninteresting field for further study. One additional measure-ment such as near-infrared spectrum analysis or moisturecontent measurements of wood entering the digester couldimprove control by allowing the model to use correct initialvalues for the model rate equations.If the controller can obtain optimised parameters for differentspecies of wood, these parameters could be stored for recallwhen switching wood species.Control can be improved considerably by removing the con-straint that the cook time needs to be constant. The controlalgorithm can be easily adapted to supply an optimal time for aconstant temperature profile instead of an optimal temperaturefor a constant time cook.In order for variable time cooks to be possible, schedulingon the plant has to be reworked. The current constant timeconstraint is due to the difficulty of manually scheduling thedigesters taking into account shared equipment and utilities.If the scheduling could be automated, the constant time con-straint could be relaxed.

eferences

agberg, B., & Schoon, N. H. (1973). Kinetical aspects of the acid sulfite cookingprocess. Part 1: Rates of dissolution of lignin and hemicellulose. SvenskPapperstidning, 76(15), 127–130.

ilian, A., & de Vaal, P. L. (2000). Potential for improved control of anacid suphite batch digester using a fundamental model. TAPPI Journal, 83(11).

arr, S. Y., & Bondy, W. B. (1986). Application of viscosity prediction modelin sulphite pulping. In Proceedings of the international sulphite conference[CPPA/TAPPI].

androck, C., & de Vaal, P. L. (2003). Performance assessment of an improvedmodel-based controller on a batch pulp digester. In Proceedings of theSAIChE national meeting [SAIChE].

erwiesch, P., & Agarwal, M. (1995). A discretized non-linear state estimatorfor batch processes. Computers and Chemical Engineering, 19(2), 238–258.

atson, E. (1992). Mathematical modeling and experimental study of the kinet-

Durban: Department of Chemical Engineering, University of Natal.orston, F. H., & Liebergott, N. (1965). Correlation of the rate of sulphite pulping

with temperature and pressure. Pulp and Paper Magazine of Canada, 66(5),272.